AlphaProof
In plain wordsA Google DeepMind system that learned to write computer-checked mathematical proofs by trial and error. In 2024 it helped an AI reach silver-medal level at the International Mathematical Olympiad for the first time.
A reinforcement learningreinforcement learning, n.: A way of teaching a computer program by trial and error: it tries actions, earns points for good results and penalties for bad ones, and gradually learns which choices pay off over time. system from Google DeepMindGoogle DeepMind, prop. n.: Google's main artificial intelligence laboratory, formed in 2023 by joining DeepMind and Google Brain. It builds the Gemini models and scientific AI systems such as AlphaFold. that searches for proofs written in Lean, a formal language whose checker verifies every step mechanically. A language model pretrained on code and mathematics learned to propose proof steps in an AlphaZeroAlphaZero, prop. n.: A 2017 DeepMind program that taught itself chess, shogi and Go from nothing but the rules by playing against itself, and within hours of training beat the strongest existing programs.-style cycle of search and learning, working on about 80 million statements obtained by machine-translating human-written competition problems into Lean, and learning from each proof or disproof the checker accepted. While tackling a hard problem it could also train on variants of that problem that it generated itself. At the 2024 International Mathematical Olympiad (IMO) it settled three problems — two in algebra, among them the contest's hardest, and one in number theory — and AlphaGeometryAlphaGeometry, prop. n.: A Google DeepMind system from 2024 that solves olympiad geometry problems nearly as well as top human competitors, by pairing a language model that suggests constructions with a logic engine that checks every step. 2 solved the geometry problem, for a combined 28 of 42 points: silver-medal level, a single point below gold. Humans had translated the problems into Lean beforehand, and some solutions took up to three days, against the nine hours allowed to contestants.
AlphaProof's proofs were ugly but airtight: Lean accepted every line.
Origin
Announced by Google DeepMind on 25 July 2024 and described in Nature in November 2025 in 'Olympiad-level formal mathematical reasoning with reinforcement learning' by Thomas Hubert, Rishi Mehta, Laurent Sartran and colleagues.
Usage note
In July 2025 an advanced Gemini Deep Think model, working in natural language within the time limits, achieved gold-medal standard (35 of 42) at the IMO, as did an experimental OpenAI model — a rapid shift from formal to natural-language mathematical reasoning.
Related
- AlphaGeometryAlphaGeometry, prop. n.: A Google DeepMind system from 2024 that solves olympiad geometry problems nearly as well as top human competitors, by pairing a language model that suggests constructions with a logic engine that checks every step.
- automated theorem provingautomated theorem proving, n.: Getting computers to prove mathematical or logical statements by themselves, by searching for a chain of valid steps that leads from basic assumptions to the statement.
- AlphaZeroAlphaZero, prop. n.: A 2017 DeepMind program that taught itself chess, shogi and Go from nothing but the rules by playing against itself, and within hours of training beat the strongest existing programs.
- reinforcement learningreinforcement learning, n.: A way of teaching a computer program by trial and error: it tries actions, earns points for good results and penalties for bad ones, and gradually learns which choices pay off over time.
- GeminiGemini, prop. n.: Google's family of AI models and its chatbot assistant, launched in December 2023. Gemini models handle text, images, audio and video and power many Google products.
- neuro-symbolic AIneuro-symbolic AI, n.: AI that combines neural networks, which learn patterns from data, with symbolic methods, which follow explicit rules and logic, aiming for systems that both learn flexibly and reason reliably.
Sources
- Hubert, T., Mehta, R., Sartran, L. et al. (2025). Olympiad-level Formal Mathematical Reasoning with Reinforcement Learning. Nature.
- AlphaProof and AlphaGeometry teams (2024). AI Achieves Silver-Medal Standard Solving International Mathematical Olympiad Problems. Google DeepMind blog, 25 July 2024. deepmind.google/discover/blog/ai-solves-imo-problems-at-silver-medal-level
Mentioned in
GeminiGemini, prop. n.: Google's family of AI models and its chatbot assistant, launched in December 2023. Gemini models handle text, images, audio and video and power many Google products., Google DeepMindGoogle DeepMind, prop. n.: Google's main artificial intelligence laboratory, formed in 2023 by joining DeepMind and Google Brain. It builds the Gemini models and scientific AI systems such as AlphaFold., automated theorem provingautomated theorem proving, n.: Getting computers to prove mathematical or logical statements by themselves, by searching for a chain of valid steps that leads from basic assumptions to the statement., International Mathematical OlympiadInternational Mathematical Olympiad, prop. n.: The world's top math competition for high-school students. Because its problems demand creative written proofs, not just answers, earning a gold medal there became a famous test of AI reasoning, first passed in 2025., LeanLean, prop. n.: A programming language and proof checker in which mathematicians write proofs that the computer verifies line by line. It has become the main language in which AI systems produce checkable proofs., Move 37Move 37, prop. n.: A surprising move played by the AI program AlphaGo against champion Lee Sedol in 2016. Experts first thought it a mistake; it proved brilliant, and the phrase now stands for a creative, unhuman idea from AI., neuro-symbolic AIneuro-symbolic AI, n.: AI that combines neural networks, which learn patterns from data, with symbolic methods, which follow explicit rules and logic, aiming for systems that both learn flexibly and reason reliably., AlphaGeometryAlphaGeometry, prop. n.: A Google DeepMind system from 2024 that solves olympiad geometry problems nearly as well as top human competitors, by pairing a language model that suggests constructions with a logic engine that checks every step., autoformalizationautoformalization, n.: Using AI to translate ordinary mathematics written for people — definitions, theorem statements, proofs — into a strict formal language that a proof-checking program can verify., MathlibMathlib, prop. n.: A huge shared library of mathematics written in the Lean language, in which every definition and proof has been checked by computer. Mathematicians keep extending it, and AI proof systems are trained on it., miniF2FminiF2F, prop. n.: A test of 488 math problems, mostly from competitions, written in a form a computer can check, used to measure how well AI systems produce proofs that are verified automatically.